Since you started with the pen, so shall I. A ball point pen used to be one of my weapons of choice: there is the personal challenge of getting it right the first time (no erasures), the inks are pigment-based and archivally permanent (and most decent sketchbooks use acid-free, buffered paper), and old-school ball point pens respond well to pressure with a variety of line widths and ink density. The more expensive pens are no better than the cheap ones, and actually tend to be blotchier in an attempt to be "more reliable". (I found that the old, white-hex-barrelled BIC fine points were the apex of technology for ball point drawing. My technique was a random scribble that looked like a mezzotint print when finished.) And perhaps 20% of pens don't need to be cleaned of excess ink around the ball after almost every stroke, so buying the 10-packs and testing is a good way to find "winners". So your construction worker may have put a great deal of effort into finding the tools he was using, and may be using $100/tube oils on Belgian linen canvas or W&N Series 7 Kolinsky sable watercolour brushes and a good 300lb+ paper at $20/sheet at home. You just don't know. And for a beginner learning classic techniques, cheap oil paints and cheap watercolour brushes are inadequate, so you have to learn all over again when you step up to the "real thing".† So put the cost factor out of your head; it's irrelevant here.
Now, about that camera...
Buying a D600 won't make you a great photographer any more than buying a Steinway will make you a great pianist. But as with the Steinway, the notes you do get right will sound better, and that may encourage you to become worthy of the instrument, so to speak.
Let's put aside the mere perks (better, faster autofocus, higher frame rate, and so forth) and concentrate on the stuff that would actually be meaningful to you. The D600 has significantly less noise at any given ISO setting than the D80, and is capable of capturing a much greater dynamic range. Pictures that would be heroic post-processing resue operations when taken with the D80 become almost easy with the D600. The top "normal" speed on the D80, ISO 1600, is just an everyday setting on the D600 (and is about as noisy as ISO 400 on your D80), and you get a couple of stops of extra, truly usable speed, along with having enough information in the RAW file to do in one exposure what would have taken at least two (and HDR-type processing) with the D80. So you can take pictures that would be practically impossible with your current camera. (Of course, much the same could be said of the $1000 D7000, but it misses the mark elsewhere.)
So the "raw material" is going to be better, but it's also bigger. You get around half again as many pixels in either direction to play with. Now, without trying to encourage sloppiness on your part, that means that you have the ability to play with the crop ex post facto and still have a usable, printable image when all's said and done. If there is one common problem that can be pointed to in most "average" photography, it's that too much was included. As Robert Capa said: "If your photographs aren't good enough, you're not close enough." Yes, the D600 will let you find good—and even great—pictures in the photos that you are taking if you take the time to critically review and mercilessly crop. And you can choose to leave it at that—or you can learn to "see" better as you review, paying close attention to what works and what doesn't. Eventually your composition in-camera will get better. And you still have plenty of room to crop when a 2:3 aspect ratio isn't right, and a squarer or more "panoramic" image suits the subject better. And if you need to retouch, more pixels is always better (even if your computer whines and complains about the processing load). (The D3200 has nearly the same pixel count, but at the cost of low-noise performance, dynamic range and handling.)
I know what my choice would be had I the two grand(ish)‡, but I can't make your decision for you. The D600 can rekindle the fire, but only if there's an ember remaining—and you actually want to rekindle it. The worst-case scenario is that you'll wind up with cleaner snapshots and mild buyer's remorse. (And you may find yourself somewhat less embarrassed about carrying and using your camera. Let's face it—when these digital thingies get long in the tooth, we do tend to become a little bit self-conscious about the highly visible model number on the camera's front face.)
† With oils, the paint is the most important component. Cheap "student" paints use substitute look-alike pigments that are often a mix of other colours, and they don't mix the same way as the good stuff (they'll produce visibly different colours when mixed, even if they look more-or-less the same out of the tube). And where expensive pigments are used, it's usually in low concentrations with inert pigments and waxy fillers taking up the space and lending body to the paint. With watercolours, the brushes and papers are key. Even the best synthetic brush will tell you nothing about the behaviour of a good Kolinsky, and the surface of a good paper has completely different absorbency, hold-back and working characteristics (for scrubbing, blotting, and techniques like sgraffito) from a "student" paper.
‡ In fact, I'd be glad to have the D80 at this point. My "budget" (for lack of a better word) barely extends to occasionally renting (or, thankfully, borrowing) decent camera bodies; the things I actually own and can use on my own terms are D70s. With a little pain, I can put aside $100/month for photographic purposes, and that has tended to go on ancillary goods like tripod, light stands, flashes, light modifiers and software as I need them.
